Eur. Phys. J. C (2011) 71: 1521DOI 10.1140/epjc/s10052-010-1521-1

Regular Article - Theoretical Physics

Model independent analysis of the forward–backward asymmetryfor the B → K1μ

+μ− decay

Ishtiaq Ahmed1,a, M. Ali Paracha2,b, M. Jamil Aslam2,c

1National Centre for Physics, Quaid-i-Azam University, Islamabad, Pakistan2Physics Department, Quaid-i-Azam University, Islambad, Pakistan

Received: 28 July 2010 / Revised: 13 November 2010 / Published online: 5 January 2011© Springer-Verlag / Società Italiana di Fisica 2011

Abstract The sensitivity of the zero position of the forward–backward asymmetry AFB for the exclusive B →K1(1270)μ+μ− decay is examined by using most generalnon-standard four-fermion interactions. Our analysis showsthat the zero position of the forward–backward asymmetryis very sensitive to the sign and size of the Wilson coef-ficients corresponding to the new vector type interactions,which are the counter partners of the usual Standard Modeloperators but have opposite chirality. In addition to these,the other significant effect comes from the interference ofscalar–pseudoscalar and tensor type operators. These resultswill not only enhance our theoretical understanding of theaxial vector mesons but will also serve as a good tool to lookfor physics beyond the SM.

1 Introduction

Flavor changing neutral current transitions (FCNC) whichgenerally arise at loop level provides a good testing groundfor the Standard Model (SM) [1–3]. Moreover, in such tran-sitions the New Physics (NP) effects can be probed via theloop of the particles that are beyond the spectrum of SM.Therefore, there are solid reasons, both theoretical and ex-perimental, for studying these FCNC transitions. Among allthe FCNC processes, the rare B decays are important sinceone can test both the SM and the possible NP effects bycomparing the theoretical results with the current and futureexperiments.

Some of the radiative and semileptonic decays of B

mesons to vector and axial vector mesons, such as B →K∗γ [4–6], B → K1(1270,1400)γ [7] and B →

a e-mail: ishtiaq@ncp.edu.pkb e-mail: ali@ncp.edu.pkc e-mail: jamil@ncp.edu.pk

K∗(892)e+e−(μ+μ−) [8, 9] have been observed and forB → K∗(892)e+e−(μ+μ−) the measurement of isospinand forward–backward asymmetry at BABAR is also re-ported [10–12]. For B → K1(1270,1400)γ Belle has giventhe following branching fractions:

Br(B → K1(1270)γ

) = (4.28 ± 0.94 ± 0.43) × 10−5,(1)

Br(B → K1(1400)γ

)< 1.44 × 10−5.

The semileptonic B meson decays, B → (K,K∗)l+l−(l = e,μ, τ ) are widely studied in the literature [13–40]where different physical observables like decay rate, leptonforward–backward asymmetry and lepton polarizations arecalculated both in SM and beyond. Among these physicalobservables, the most interesting one is the lepton forward–backward asymmetry AFB and this lies in the vanishing ofAFB at a specific value of dilepton mass in a hadronicallyclean way [41–44]. This in other words provides a simple re-lationship between the electric dipole coefficient C7 and C9,which is almost free from the hadronic uncertainties whicharises dominantly from the form factors [42].

The above mentioned decays also open a window to lookfor new physics. We know that in SM the decays B →(K,K∗)l+l− are completely determined by the Wilson co-efficients of only three operators O7, O9 and O10 which areevaluated at the scale μ = mb [45–47]. On the other handthe most general analysis of these decays needs other set ofnew operators which are based on the general four-fermioninteractions. The new structure of effective Hamiltonian [48,49] makes them an ideal platform for the SM, and providesclues for the NP. In the literature, the model independentanalysis of the quark level b → sl+l− decay, in terms of 10new types of local four-fermion interactions, has been per-formed in [48] which is then applied to the systematic studyof B → (K,K∗)l+l− [50, 51]. Recently, the discrepancyhas been observed in the lepton forward–backward asym-metry in the exclusive B → K∗μ+μ− decay [52–55]. To

Page 2 of 10 Eur. Phys. J. C (2011) 71: 1521

explain the experimental results, Kumar et al. [56] have donea systematic study B → K∗μ+μ− decay by using the mostgeneral model independent Hamiltonian. They have shownthat though the scalar and tensor operators are not very im-portant to study the lepton forward–backward asymmetrythe interference of these two is important and is not ignor-able, which differs from the results given in [50, 51].

As the radiative decay B → K1(1270)γ has already beenseen by Belle, the related decay with a lepton pair instead ofa photon in the final state can also be expected to be seen.Analysis of this decay process will be a useful complementto the widely investigated analysis for the B → K∗l+l−process, since the analysis probes the effective Hamiltonianin a similar but not identical way. The experimental investi-gation of this decay will thus provide us with an independenttest of the predication of the SM and also give us a clue forNP.

Like B → K∗l+l− the semileptonic decay B →K1(1270)l+l− is also governed by the quark level transi-tion b → sl+l−. Compared to B → K∗l+l− the situation iscomplicated in the decay B → K1(1270)l+l−, because theaxial vector states K1(1270) and K1(1400) are mixtures ofideal 1P1(K1A) and 3P1(K1B) orbital angular momentumstates and the current limit on the mixing angle is [57]

θ = −(34 ± 13)◦. (2)

Recently, some studies have been made on B → K1 transi-tions both by incorporating the mixing angle as well as without it [58–69].

Experimentally, this decay has not yet been seen, but itis expected to be observed at LHC [70, 71] and SuperB fac-tory [72–75]. In particular, the LHCb experiment at the LHCwhere estimates made in [70, 71, 76] for LHCb collabora-tion show that with an integrated luminosity of 2 fb−1, onemay expect almost 8000 B → K∗l+l− events. Although thebranching ratio of B → K1(1270)l+l− calculated in [77] isan order of magnitude smaller than the experimentally mea-sured value of B → K∗l+l− [78], but still one can expectthe significant number of events for this decay and hencemaking an analysis of FB asymmetry for this decay will beexperimentally meaningful for comparison with the SM andthe theories beyond it.

In this work, our aim is to analyze the possible newphysics effects stemming from the new structures in the ef-fective Hamiltonian [49] to the forward–backward asymme-try for the B → K1(1270)l+l− decay. It has already beenmentioned that some experimental analysis for the decayB → K∗μ+μ− has already been studied in B factories [72–75], but only the large increase in statistics at LHCb forB → K∗μ+μ− will make much higher precision measure-ments possible [70, 71, 76]. It is known that the forward–backward asymmetry becomes zero for a particular valueof the dilepton invariant mass. In the SM, the zero of the

AFB(q2) appears in the low q2 region, sufficiently awayfrom the charm resonance region and is almost free fromthe hadronic uncertainties (i.e. the choice of form factors)and so is from the mixing angle. Now this zero position ofAFB varies from model to model and this makes it an im-portant tool to search for physics beyond the SM. The orga-nization of the paper is as follows: In Sect. 2 we introducethe model independent effective Hamiltonian and obtain thetransition matrix elements in terms of form factors of theB → K1(1270)l+l−. Section 3 describes the formulas thatcan be used to determine the zero position of the FBA. InSect. 4 we present our numerical analysis and Sect. 5 sum-marizes our conclusion.

2 Effective Hamiltonian and matrix elements

By integrating out the heavy degrees of freedom in the fulltheory, the general effective Hamiltonian for b → sl+l−transitions in the SM can be written as

Heff = −4GF√2

VtbV∗ts

[10∑

i=1

Ci(μ)Oi(μ)

]

, (3)

where Oi(μ) (i = 1, . . . ,10) are the four-quark operatorsand Ci(μ) are the corresponding Wilson coefficients at theenergy scale μ [45–47]. Using renormalization group equa-tions to resum the QCD corrections, Wilson coefficientsare evaluated at the energy scale μ = mb . The theoreticaluncertainties associated with the renormalization scale canbe substantially reduced when the next-to-leading-logarithmcorrections are included.

The explicit expressions of the operators responsible forexclusive B → K1(1270)l+l− transition are given by

O7 = e2

16π2mb(sσμνPRb)Fμν, (4)

O9 = e2

16π2(sγμPLb)

(lγ μl

), (5)

O10 = e2

16π2(sγμPLb)

(lγ μγ5l

), (6)

with PL,R = (1 ± γ5)/2. In terms of the above Hamiltonian,the free quark decay amplitude for b → s l+l− is

MSM(b → sl+l−

)

= −GF α√2π

VtbV∗ts

{Ceff

9 (sγμPLb)(lγ μl

)

+ C10(sγμPLb)(lγ μγ5l

)

− 2mbCeff7

(siσμν

qν

sPRb

)(lγ μl

)}, (7)

Eur. Phys. J. C (2011) 71: 1521 Page 3 of 10

where s = q2 and q is the momentum transfer. The operatorO10 cannot be induced by the insertion of four-quark oper-ators because of the absence of the Z boson in the effectivetheory. Therefore, the Wilson coefficient C10 does not renor-malize under QCD corrections and hence it is independentof the energy scale. In addition to this, the above quark leveldecay amplitude can receive contributions from the matrixelement of four-quark operators,

∑6i=1〈l+l−s|Oi |b〉, which

are usually absorbed into the effective Wilson coefficientCeff

9 (μ), which one can decompose into the following threeparts [79–85]:

Ceff9 (μ) = C9(μ) + YSD(z, s′) + YLD(z, s′),

where the parameters z and s′ are defined as z = mc/mb ,s′ = q2/m2

b . YSD(z, s′) describes the short-distance contri-butions from four-quark operators far away from the cc res-onance regions, which can be calculated reliably in the per-turbative theory. The long-distance contributions YLD(z, s′)from four-quark operators near the cc resonance cannot becalculated from first principles of QCD and are usually para-meterized in the form of a phenomenological Breit–Wignerformula making use of the vacuum saturation approxima-tion and quark–hadron duality. The manifest expressions forYSD(z, s′) and YLD(z, s′) can be written as

YSD(z, s′) = h(z, s′)(3C1(μ) + C2(μ)

+ 3C3(μ) + C4(μ) + 3C5(μ) + C6(μ))

− 1

2h(1, s′)

(4C3(μ) + 4C4(μ)

+ 3C5(μ) + C6(μ))

− 1

2h(0, s′)

(C3(μ) + 3C4(μ)

)

+ 2

9

(3C3(μ) + C4(μ) + 3C5(μ) + C6(μ)

),

(8)

with

h(z, s′) = −8

9ln z + 8

27+ 4

9x − 2

9(2 + x)|1 − x|1/2

×⎧⎨

⎩ln

∣∣∣√

1−x+1√1−x−1

∣∣∣ − iπ for x ≡ 4z2/s′ < 1,

2 arctan 1√x−1

for x ≡ 4z2/s′ > 1,

h(0, s′) = 8

27− 8

9ln

mb

μ− 4

9ln s′ + 4

9iπ (9)

and

YLD(z, s′) = 3π

α2C(0)

∑

Vi=ψi

κi

mViΓ (Vi → l+l−)

m2Vi

− s′m2b − imVi

ΓVi

(10)

where C(0) = 3C1 + C2 + 3C3 + C4 + 3C5 + C6. TheYLD(z, s′) critically depend on the resonance model usedto describe these LD contributions and as such they haveuncertainties. But these uncertainties will hardly affect thezero position of the FB asymmetry, which lies below thischarmonium threshold. Keeping in view that there are no ex-perimental data on B → K1(1270)l+l−, we have fixed thevalues of the phenomenological parameters κi from B →K∗l+l−, which for the resonances J/Ψ and Ψ

′are taken to

be κ = 1.65 and κ = 2.36, respectively [42].Apart from this, the non-factorizable effects [86–89]

from the charm loop can bring about further corrections tothe radiative b → sγ transition, which can be absorbed intothe effective Wilson coefficient Ceff

7 . Specifically, the Wil-son coefficient Ceff

7 is given by [90–93]

Ceff7 (μ) = C7(μ) + Cb→sγ (μ),

with

Cb→sγ (μ)

= iαs

[2

9η14/23(G1(xt ) − 0.1687

) − 0.03C2(μ)

], (11)

G1(x) = x(x2 − 5x − 2)

8(x − 1)3+ 3x2ln2x

4(x − 1)4, (12)

where η = αs(mW)/αs(μ), xt = m2t /m2

W , Cb→sγ is the ab-sorptive part for the b → scc → sγ rescattering and we havedropped the tiny contributions proportional to CKM sectorVubV

∗us.

In addition to the above mentioned currents, the mostgeneral form of the effective Hamiltonian contains 10 lo-cal four-fermion interactions which can contribute to theB → K1(1270)l+l− decay and these can be written as

Mnew(b → sl+l−

) = MV −A + MS−P + MT ,

MV −A = GF α√2π

V ∗tsVtb

{CLLsLγ μbLlLγ μlL

+ CLRsLγ μbLlRγ μlR

+ CRLsRγ μbRlLγ μlL

+ CRRsRγ μbRlRγ μlR},

MS−P = GF α√2π

V ∗tsVtbCLRLRsLbRlLlR

+ CRLLRsRbLlLlR

+ CRLRLsRbLlRlL,

MT = GF α√2π

V ∗tsVtb

{CT sσμνblσμνl

+ iCTEεμναβlσμνlsσαβb}.

(13)

Page 4 of 10 Eur. Phys. J. C (2011) 71: 1521

Thus the explicit form of the free quark amplitude M forthe b → sl+l− transition can be written as sum of the SMamplitude (7) and of the new physics contributions (13),i.e.

M = MSM + Mnew. (14)

The exclusive B → K1(1270)l+l− decay involves thehadronic matrix elements of quark operators given in (7)and (13), which one may parametrize in terms of the formfactors as follows:

⟨K1(k, ε)

∣∣Vμ

∣∣B(p)⟩

= ε∗μ(MB + MK1)V1(s)

− (p + k)μ(ε∗ · q)V2(s)

MB + MK1

− qμ(ε · q)2MK1

s

[V3(s) − V0(s)

], (15)

⟨K1(k, ε)

∣∣Aμ

∣∣B(p)⟩ = 2iεμναβ

MB + MK1

ε∗νpαkβA(s), (16)

where Vμ = sγμb and Aμ = sγμγ5b are the vectors andaxial vector currents, respectively. Also p(k) are the mo-mentum of the B(K1) meson and ε∗

μ is the polariza-tion of the final state axial vector K1 meson. In (15) wehave

V3(s) = MB + MK1

2MK1

V1(s) − MB − MK1

2MK1

V2(s), (17)

with

V3(0) = V0(0).

In addition to the above, there is also a contribution from thepenguin form factors, which can be written as

⟨K1(k, ε)

∣∣siσμνqνb

∣∣B(p)⟩

= [(M2

B − M2K1

)εμ − (ε · q)(p + k)μ

]F2(s)

+ (ε∗ · q)

[qμ − s

M2B − M2

K1

(p + k)μ

]F3(s), (18)

⟨K1(k, ε)

∣∣siσμνqνγ5b

∣∣B(p)⟩

= −iεμναβε∗νpαkβF1(s), (19)

with F1(0) = 2F2(0).By contracting (15) with qμ and making use of the equa-

tion of motions

qμ(ψ1γμψ2) = (m2 − m1)ψ1ψ2, (20)

qμ(ψ1γμγ5ψ2) = −(m1 + m2)ψ1γ5ψ2, (21)

we have

⟨K1(k, ε)

∣∣s(1 ± γ5)b∣∣B(p)

⟩

= 1

mb + ms

{∓2iMK1(ε∗ · q)V0(s)

}. (22)

The form factors for B → K1(1270) transition are non-perturbative quantities and need to be calculated using dif-ferent approaches (both perturbative and non-perturbative)like lattice QCD, QCD sum rules, light cone sum rules,etc. As the zero position of the forward–backward asym-metry depends on the short-distance contribution i.e. theWilson coefficients; it is not very sensitive to the long-distance contribution (form factors) [77] and consequentlyon the mixing angle between 1P1 and 3P1 states. As suchwe will consider the form factors that were calculated us-ing Ward Identities in [77] which can be summarized as fol-lows:

A(s) = A(0)

(1 − s)

(1 − s

M2B

M ′2B

),

V1(s) = V1(0)(1 − s

M2B

M2B∗

A

)(1 − s

M2B

M ′2B∗

A

)

(1 − s

1 − M2K1

),

(23)

V2(s) = V2(0)(1 − s

M2B

M2B∗

A

)(1 − s

M2B

M ′2B∗

A

)

− 2MK1

1 − MK1

V0(0)

(1 − s)(1 − s

M2B

M ′2B

) ,

with

V0(0) = 0.36 ± 0.03, (24)

A(0) = −(0.52 ± 0.05),

V1(0) = −(0.24 ± 0.02), (25)

V2(0) = −(0.39 ± 0.05).

3 Forward–backward asymmetryfor B → K1(1270)l+l−

In this section, we are going to perform the calculation of theforward–backward asymmetry. From (7), it is straightfor-ward to obtain the decay amplitude for B → K1(1270)l+l−as

MB→K1(1270)l+l−

= GF α

4√

2πVtbV

∗tsMB

{T 1

μlγ μl + T 2μlγ μγ 5l + T 3ll

Eur. Phys. J. C (2011) 71: 1521 Page 5 of 10

+ T 4lγ 5l + 8CT

(lσμνl

)(−2F1(s)ε∗μ(pB + pK1)

μ

+ J1ε∗μqν − J2

(ε∗ · q)

pμK1

qν)

+ 2iCTEεμναβ

(lσμνl

)(−2F1(s)ε∗α(pB + pK1)

β

+ J1ε∗αqβ − J2(ε

∗ · q)pαK1

qβ)}

(26)

where the functions T 1μ , T 2

μ , T 3 and T 4 in terms of auxiliaryfunctions are given by

T 1μ = iA′(s)εμραβε∗ρpα

BpβK1

− B ′(s)ε∗μ

+ C′(s)(ε∗ · pB)phμ + D′(s)(ε∗ · pB)qμ,

T 2μ = iE′(s)εμραβε∗ρpα

BpβK1

− F ′(s)ε∗μ

(27)+ G′(s)(ε∗ · pB)phμ + H ′(s)(ε∗ · pB)qμ,

T 3 = iI ′(ε∗ · q),

T 4 = iJ ′(ε∗ · q),

where s = s/M2B , pK1 = pK1/MB , pB = pB/MB , mb =

mb/MB and MK1 = MK1/MB .Defining the combinations

C(+)RR = CRR + CRL, C

(−)RR = CRR − CRL,

C(+)LL = CLL + CLR, C

(−)LL = CLL − CLR,

(28)C

(+)RLLR = CRLLR + CRLRL, C

(+)LRRL = CLRRL + CLRLR,

C(−)RLLR = CRLLR − CRLRL, C

(−)LRRL = CLRLR − CLRRL,

the auxiliary functions appearing in (27) can be written asfollows:

A′(s) = − 2

1 + MK1

[Ceff

9 + 1

2

(C

(+)RR + C

(+)LL

)]A(s)

+ 2mb

sCeff

7 F1(s),

B ′(s) = (1 + MK1)

(Ceff

9 + 1

2

(C

(+)LL − C

(−)RR

))

V1(s)

+ 2mb

s

(1 − M2

K1

)Ceff

7 F2(s),

C′(s) = 1

(1 − M2K1

)

[((1 − MK1)

×(

Ceff9 + 1

2

(C

(+)LL − C

(+)RR

)))

V2(s)

+ 2mbCeff7

(F3(s) − (

1 − M2K1

)/s

)F2(s)

],

D′(s) = 1

s

[((1 + MK1)V1(s)

− (1 − MK1)V2(s) − 2MK1V0(s))

×(

Ceff9 + 1

2

(C

(+)LL − C

(+)RR

))

(29)

− 2mbCeff7 F3(s)

],

E′(s) = −2

1 + MK1

[C10 + 1

2

(C

(−)RR − C

(−)LL

)]A(s),

F ′(s) = (1 + MK1)

[C10 − 1

2

(C

(−)LL + C

(−)RR

)]V1(s),

G′(s) = − 1

(1 + MK1)

[C10 − 1

2

(C

(−)LL + C

(−)RR

)]V2(s),

H ′(s) = 1

s

[((1 − MK1)V2(s) − (1 + MK1)V1(s)

+ 2MK1V0(s))(

C10 − 1

2

(C

(−)RR + C

(−)LL

))],

I′(s) = 2MK1

mb

V0(s)[C

(+)RLLR + C

(+)LRRL

],

J′(s) = 2MK1

mb

V0(s)[C

(+)RLLR − C

(+)LRRL

],

J ′1(s) = 2

(1 − M2

K1

)F1(s) − F2(s)

s,

J ′2(s) = 4M2

B

s

(F1(s) − F2(s) − s

1 − M2K1

F3(s)

),

where A′, B ′, C′, D′, E′, F ′, G′, H ′ correspond to V A in-teractions, whereas I ′, J ′ , J ′

1, J ′2 are relevant for SP and T

interactions.To calculate the forward–backward asymmetry of the fi-

nal state leptons, one needs to know the differential decaywidth of B → K1(1270)l+l−, which in the rest frame of B

meson can be written as

dΓ (B → K1(1270)l+l−)

ds

= 1

(2π)3

1

32MB

∫ umax

umin

|MB→K1(1270)l+l−|2 du, (30)

where u = (k + pl−)2 and s = (pl+ + pl−)2; k, pl+ and pl−are the four-momenta vectors of K1(1270), l+ and l−, re-spectively; |MB→K1(1270)l+l−|2 is the squared decay ampli-tude after integrating over the angle between the lepton l−and K1(1270) meson. The upper and lower limits of u are

Page 6 of 10 Eur. Phys. J. C (2011) 71: 1521

given by

umax = (E∗K1(1270) + E∗

l−)2

− (√E∗2

K1(1270) − M2K1(1270) −

√E∗2

l− − m2l−

)2,

(31)umin = (E∗

K1(1270) + E∗l−)2

− (√E∗2

K1(1270) − M2K1(1270) +

√E∗2

l− − m2l−

)2,

where E∗K1(1270)

and E∗l− are the energies of K1(1270) and

l− in the rest frame of lepton pair and can be determined as

E∗K1(1270) = M2

B − M2K1(1270) − s

2√

s,

E∗l = s

2√

s.

(32)

The differential FBA of the final state lepton for the saiddecay can be written as

dAFB(s)

ds=

∫ 1

0d cos θ

d2Γ (s, cos θ)

ds d cos θ

−∫ 0

−1d cos θ

d2Γ (s, cos θ)

ds d cos θ(33)

and

AFB(s) =∫ 1

0 d cos θd2Γ (s,cos θ)

ds d cos θ− ∫ 0

−1 d cos θd2Γ (s,cos θ)

ds d cos θ∫ 1

0 d cos θd2Γ (s,cos θ)

ds d cos θ+ ∫ 0

−1 d cos θd2Γ (s,cos θ)

ds d cos θ

.

(34)

Now putting everything together in hat notation we have

dAFB

ds= G2

F α2m5B

210π5|V ∗

tsVtb|2u(s)[XVA + XSP + XT

+ XVA−SP + XVA−T + XSP−T ], (35)

where

u(s) =√

λ(1, ˆMK1 , s)

(1 − 4

m2l

s

)

λ(1, M2

K1, s

) = 1 + M4K1

+ s2 − 2s − 2M2K1

(1 + s)

and

XVA = MBsMK1�[A′∗F ′ + B ′∗E′],XSP = 0,

XT = 0,

XSP−VA = ml

[(M2

K1+ s − 1

)�(B ′∗I ′)

+ M2Bλ�(I ′∗C′)

],

XSP−T = MBM2K1

�[2I ′∗CT + J ′∗CTE)

× (2J ′

1

(M2

K1+ s − 1

) + J ′2M

2Bλ

+ 4F1(s)(3M2

K1− s + 1

)),

XVA−T = ml

[2�(F ′∗CTE)

(2J ′

1

(M2

K1+ s − 1

)(36)

+ J ′2M

2Bλ + F1(s)

(4M2

K1− 4s + 4

))

− 2�(G′∗CTE)M2B

(2J ′

1

(M2

K1s − s2 + s + λ

)

+ J ′2M

2B

(M2

K1− 1

)λ

+ 4F1(s)(5M2

K1s + 4M2

K1− 3s2 + 7s + 3λ − 4

))

+ 2�(H ′∗CTE)M2BM2

K1

(2J ′

1

(M2

K1+ s − 1

)

+ J ′2M

2Bλ + 4F1(s)

(3M2

K1− s + 1

))

− 64�(E′∗CT )M2B

(J1M

2K1

s

+ 2F1(s)(M2

K1s + s − (s − 1)2 + λ

))].

From the experimental point of view the normalized forward–backward asymmetry is more useful, i.e.

dAFB

ds=

dAFBdsdΓds

.

4 Numerical analysis

In the following section, we examine the lepton forward–backward asymmetry and study the sensitivity of its zeroposition to new physics operators. We consider differentLorentz structures of NP, as well as their combinations andtake all the NP couplings to be real.

4.1 Switching off all new physics operators

By switching off all the new physics operators one will getthe SM result of the lepton forward–backward asymmetryfor B → K1(1270)μ+μ−, which was earlier calculated byParacha et al. [77] and has been shown by solid lines in allthe figures shown below. The zero position lies at s = 0.16(s = 4.46 GeV−2) and is almost independent of the choiceof form factors and also from the uncertainties arising fromdifferent input parameters like form factors, CKM matrix el-ements, etc. In the subsequent analysis we will ignore theseuncertainties.

In case of B → K∗, Arda et al. have shown [94] that thepresence of the tensor and the scalar type interactions havea very mild effect on the zero position of forward–backwardasymmetry (AFB) and they have ignored it in their analy-sis. However, recently the discrepancy has been observedin the lepton forward–backward asymmetry in the exclusiveB → K∗μ+μ− decay [52–55]. To explain the experimen-

Eur. Phys. J. C (2011) 71: 1521 Page 7 of 10

(a)

(b)

Fig. 1 Forward–backward asymmetry for the B → K1μ+μ− decays

as functions of s for different values of CLL(LR). Solid line correspondto SM value, dashed line is for CLL(LR) = −C10, dashed dot dot is forCLL(LR) = −0.7C10, dashed dotted line is for CLL(LR) = C10, dashedtriple dotted is for CLL(LR) = 0.7C10. The coefficients of the other in-teractions are all set to zero

tal results, Kumar et al. [56] have done a systematic studyof B → K∗μ+μ− decay by using the most general modelindependent Hamiltonian. They have shown that though thescalar and tensor operators are not important to study thelepton forward–backward asymmetry but the interference ofthese two is important and is not ignorable. Therefore, keep-ing this in view we will not ignore these scalar and tensortype couplings in our analysis of B → K1(1270) decay. Inorder to see the effect of the new vector type Wilson co-efficients (CX = CLL,CLR,CRR,CRL,CLRLR,CT ,CTE), wehave plotted the dependence of AFB on s by using differentvalues of CX , which can be summarized as follows.

4.2 Switching on only CLL and CLR along with SMoperators

Considering the constraints provided by Kumar et al. [56]we took a broad range of the values of different VA cou-plings. Figure 1(a, b) shows the dependence of AFB on s

when all the CLL and CLR are present. When CLL(LR) =

(a)

(b)

Fig. 2 Forward–backward asymmetry for the B → K1μ+μ− decays

as functions of s for different values of CRR(RL). Solid line correspondto SM value, dashed line is for CRR(RL) = −C10, dashed dot dot is forCRR(RL) = −0.7C10, dashed dotted line is for CRR(RL) = C10, dashedtriple dotted is for CRR(RL) = 0.7C10. The coefficients of the other in-teractions are all set to zero

−C10, CLL(LR) = C10, CLL(LR) = −0.7 × C10, CLL(LR) =0.7 × C10 (and all other Wilson coefficients are set to zero)we denote the curves of AFB by dashed double dotted,dashed triple dotted, dashed and dashed dotted lines, respec-tively. The solid line corresponds to the SM result. One candeduce from this that there is a significant shift in the zeroposition of the forward–backward asymmetry and the posi-tion of zero is gradually shifted to the left for positive valuesof C10 and to the right for negative values of C10 comparedto the SM value. This is contrary to the B → K∗μ+μ− de-cay process where for the positive values of CLL(LR) thezero position of AFB shifts to the right and for negative valueof these new coefficients the shift in the zero position is tothe left [95]. This difference is due to the axial vector natureof the K1(1270). For different values of NP coefficients, thelocation of the zero of the AFB varies from s = 0.12 to 0.23.

4.3 Switching on CRR and CRL along with SM operators

In Fig. 2(a, b) we have shown the dependence of forward–backward asymmetry on CRR and CRL. Figure 2a gives the

Page 8 of 10 Eur. Phys. J. C (2011) 71: 1521

Fig. 3 Forward–backward asymmetry for the B → K1μ+μ− decays

as functions of s for different values of scalar and pseudoscalar oper-ators. Solid line corresponds to SM value, dashed line is for R = 0.44and dashed dotted is for R < 0.44. The coefficients of the other NPinteractions are all set to zero

plot of the AFB with s by using different values of CRR andsetting all the other Wilson Coefficients to zero. By varyingthe CRR from −C10 to C10 in the same way as we did forthe CLL in Fig. 1, we have plotted the AFB with s in Fig. 2awhere the legends of the curves are the same as in Fig. 1.One can clearly see that the zero position of the forward–backward asymmetry is less sensitive to CRR compared tothe CLL and CLR and the position of the zero shifts left to theSM value from s = 0.16 to 0.12 when CRR is changed from−C10 to C10. Again this is contrary to the B → K∗μ+μ−case where is the shift of zero position of AFB is the otherway.

Similarly Fig. 2b shows the dependency of the zero po-sition of forward–backward asymmetry on different valuesof CRL. It can be seen that when CRL varies from −C10

to C10, the zero position of the AFB shifts gradually rightto the SM value from s = 0.16 to 0.21.

4.4 Switching only scalar–pseudoscalar(CLRLR,CRLLR,CLRRL,CRLRL) operators along withSM operators

Figure 3 shows the behavior of the lepton forward–backwardasymmetry for different NP scalar operators. In the graphwe have chosen the value of the scalar and pseudoscalar op-erators such that they satisfy the constraint R ≡ |C(+)

LRRL −C

(+)RLLR|2 + |C(−)

LRRL − C(−)RLLR|2 ≤ 0.44 as provided by the

B0s → μ+μ− decay [56]. It can be seen from (36) that the

contribution from the scalar operators alone is zero. This isquite clear in the graph where the value of AFB overlaps withthat of the SM value and this is due to the interference be-tween the NP scalar operators and that of the SM operators(i.e. their coefficients).

Fig. 4 Forward–backward asymmetry for the B → K1μ+μ− de-

cays as functions of s for different values of scalar and pseudoscalaroperators. Solid line corresponds to SM value, dashed line is for|CT |2 + 4|CTE|2 = 1.3, dashed dot is for |CT |2 + 4|CTE|2 = 0.9. Thecoefficients of the other NP interactions are all set to zero

4.5 Switching on only tensor–axial tensor (CT ,CTE)

operators along with SM operators

This is the case where only NP tensor operators are added.It is expected from (36) that the contribution alone from thetensor operators to AFB is zero and Fig. 4 reflects this sce-nario. Just like the scalar operators, the non-zero value ofthe forward–backward asymmetry is due to the interferencebetween the tensor type operator and of the SM operatorsand these are ml suppressed (cf. (36)). The allowed valuesof new tensor type operators are restricted to be [56]

|CT |2 + 4|CTE|2 ≤ 1.3. (37)

In Fig. 4 one can see the ml suppression (which is notnegligible) for the value of AFB(s) in the low s region.Though the value is suppressed still the shift in the zero po-sition is quite significant in the low s region, which is due tothe mixing of tensor and SM interactions.

4.6 Combination of SP, VA and T operators

Apart from the individual contribution of NP operators andtheir interference with the SM operators there is also a mix-ing between NP operators by itself. By looking at the termXSP−VA in (36) one can see that it is ml suppressed but withthe second term there is a factor of M2

B which will overcomethis suppression. This will not only change the zero positionof AFB but this also increases or decreases its value com-pared to the SM value depending on the size and sign of NPcouplings. In Fig. 5, we took R = 0.44 and the values of NPvector type operators are taken to be 0.3C10 or −0.3C10.

Among different mixing terms the most important is theSP and T term. Though the individual contribution of SPand T to the AFB are not very significant their interferenceterm is quite promising. One can see from XSP−T term in

Eur. Phys. J. C (2011) 71: 1521 Page 9 of 10

Fig. 5 Forward–backward asymmetry for the B → K1μ+μ− decays

as functions of s for different values of vector and axial-vector oper-ators. Solid line corresponds to SM value, dashed line is for VA cou-plings equal to −0.3C10 and dashed dot dot lines are for VA equal to0.3C10. Here we took the value of SP operators such that they satisfyR = 0.44. The coefficients of the other NP interactions are all set tozero

Fig. 6 Forward–backward asymmetry for the B → K1μ+μ− decays

as functions of s. Solid line corresponds to SM value, dashed line is for|CT |2 +4|CTE|2 = 1.3 and dashed dot dot is for |CT |2 +4|CTE|2 = 0.9.Here we kept R = 0.44 and the coefficients of the other VA NP inter-actions are all set to zero

(36) in which there is no lepton mass suppression. In Fig. 6,we have shown the dependencies of the zero position of theforward–backward asymmetry for different values of the SPcouplings. The value of the tensor couplings is chosen to be|CT |2 + 4|CTE|2 ≤ 1.3.

Finally, the contribution from the mixing terms of VA andT is suppressed by ml which can be seen in the XVA−T termof (36).

5 Conclusion

The sensitivity of the zero position of the forward–backwardasymmetry to new physics effects is studied here. Weshowed that the position of the zero of the forward–backward asymmetry shifts significantly from its Standard

Model value both for the size and sign of the vector–vectornew physics operators which are the opposite chirality partof the corresponding SM operators. The scalar–scalar four-fermion interactions have very mild effects on the zero of theforward–backward asymmetry. The tensor type interactionsshift the zero position of the forward–backward asymmetrybut these are ml suppressed. However, the interference ofSP and T operators gives a significant change in the zeroposition of AFB.

In short, our results provide, just as in case of the B →K∗l+l− process, an opportunity for the straightforwardcomparison of the basic theory with the experimental re-sults, which may be expected in near future for this process.

Acknowledgements The authors would like to thank Profs. Riazud-din and Fayyazuddin for their valuable guidance and helpful discus-sions. The authors M.A.P. and M.J.A. would like to acknowledge thefacilities provided by National Centre for Physics during this work.

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